Structural Column Strain Energy Stability (Civil & Mechanical Engineering) Solution

A load-bearing structural steel support column in a commercial building framework experiences internal strain energy $U$ (in Joules, $\text{J}$) due to combined axial mechanical compression loads and solar thermal expansion. The total strain energy is modeled as a function of the axial force load $x$ (in kilonewtons, $\text{kN}$) and the surface temperature shift $y$ (in degrees Celsius, $^\circ\text{C}$):

1. Compute the first-order partial derivatives $U_x$ and $U_y$.
2. Compute all four second-order partial derivatives ($U_{xx}, U_{yy}, U_{xy}, U_{yx}$).
3. Verify Clairaut's Theorem for the mixed partial derivatives.
4. Construct the $2 \times 2$ Hessian matrix $\mathbf{H}_U(x, y)$ and evaluate its determinant $D = \det(\mathbf{H}_U)$ at nominal service load $(x, y) = (10\text{ kN}, 25^\circ\text{C})$.
5. Interpret the physical structural meaning of the mixed partial derivative $U_{xy}$ and explain why $U_{xx} > 0$ confirms structural stability against buckling.

Step-by-Step Solution:

1. Compute First-Order Partial Derivatives: State the First-Order Partial Derivative Definitions (see Partial Derivatives):

   $$U_x = \frac{\partial}{\partial x}[0.5x^2 + 0.04x^2 y + 3y^2] = 1.0x + 0.08xy$$
   $$U_y = \frac{\partial}{\partial y}[0.5x^2 + 0.04x^2 y + 3y^2] = 0.04x^2 + 6y$$

2. Compute Pure and Mixed Second-Order Partials: State the Second-Order Partial Derivative Formulas (see Second-Order Partial Derivatives):

   • Pure Partials:
     $$U_{xx} = \frac{\partial}{\partial x}[x + 0.08xy] = 1.0 + 0.08y$$
     $$U_{yy} = \frac{\partial}{\partial y}[0.04x^2 + 6y] = 6$$
   • Mixed Partials:
     $$U_{xy} = \frac{\partial}{\partial y}[x + 0.08xy] = 0.08x$$
     $$U_{yx} = \frac{\partial}{\partial x}[0.04x^2 + 6y] = 0.08x$$

3. Verify Clairaut's Theorem:

   $$U_{xy} = U_{yx} = 0.08x$$
   Since $U(x, y)$ is a smooth polynomial, its second partials are continuous everywhere on $\mathbb{R}^2$, verifying Clairaut's Theorem.

4. Construct Hessian Matrix and Evaluate Determinant: State the Hessian Matrix definition:

   $$\mathbf{H}_U(x, y) = \begin{bmatrix} U_{xx} & U_{xy} \\ U_{yx} & U_{yy} \end{bmatrix} = \begin{bmatrix} 1.0 + 0.08y & 0.08x \\ 0.08x & 6 \end{bmatrix}$$
   • Evaluate at $(x, y) = (10, 25)$:
     • $U_{xx}(10, 25) = 1.0 + 0.08(25) = 1.0 + 2.0 = 3.0\text{ J/kN}^2$
     • $U_{yy}(10, 25) = 6.0\text{ J/}(^\circ\text{C})^2$
     • $U_{xy}(10, 25) = 0.08(10) = 0.8\text{ J/(kN}\cdot^\circ\text{C)}$
       $$\mathbf{H}_U(10, 25) = \begin{bmatrix} 3.0 & 0.8 \\ 0.8 & 6.0 \end{bmatrix}$$
   • Determinant:
     $$D = \det(\mathbf{H}_U) = (3.0)(6.0) - (0.8)^2 = 18.0 - 0.64 = 17.36 > 0$$

5. Physical & Structural Interpretation:

   • Mixed Partial $U_{xy}$: Physically represents the thermal-mechanical coupling stiffness. It measures how thermal expansion increases the column's mechanical stiffness response to compressive loading.
   • Stability & Convexity: Because $D = 17.36 > 0$ and $U_{xx} = 3.0 > 0$, the strain energy function is strictly convex (concave upward) in all directions. In structural mechanics, positive curvature of the strain energy potential proves that the column is in a stable equilibrium state, guaranteeing resistance against sudden elastic buckling under these combined thermal-load service conditions.